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Chapter 1: Problem 9

According to Kepler's third law, the orbital period \(T\) of a planet is related to the radius \(R\) of its orbit by \(T^{2} \propto R^{3}\) Jupiter's orbit is larger than Earth's by a factor of 5.19 What is Jupiter's orbital period? (Earth's orbital period is \(1 \text { yr. })\)

Short Answer

Expert verified
Answer: Jupiter's orbital period is approximately 11.86 years.

Step by step solution

01

Write the Kepler's third law formula

Kepler's third law states that the square of the period of two planets is proportional to the cube of their average distance from the sun. In mathematical form, the relationship becomes: $$\frac{T^{2}_{1}}{T^{2}_{2}} = \frac{R^{3}_{1}}{R^{3}_{2}}.$$
02

Assign values to the variables

According to the problem, Jupiter's orbit is larger than Earth's by a factor of 5.19, so we can write \(R_{2} = 5.19R_{1}\). Earth's orbital period (\(T_{1}\)) is 1 year, and we need to find Jupiter's orbital period (\(T_{2}\)).
03

Substitute the values into the equation and solve for \(T_2\)

Substitute the known values of \(R_{1}\), \(R_{2}\), and \(T_{1}\) into the equation: $$\frac{1^2}{T^{2}_{2}} = \frac{R^{3}_{1}}{(5.19R_{1})^{3}}.$$
04

Simplify the equation

Cubing 5.19 and simplifying the expression, we get: $$\frac{1}{T^{2}_{2}} = \frac{1}{5.19^{3}}.$$ Now, our objective is to find the value of \(T^{2}_{2}\).
05

Solve for \(T_2\)

To find \(T^{2}_{2}\), we can simply invert both sides of the equation: $$T^{2}_{2} = 5.19^{3}.$$ Now, to find the value of \(T_2\), we take the square root of both sides: $$T_{2} = \sqrt{5.19^{3}}.$$
06

Calculate Jupiter's orbital period

Using a calculator, we compute the value of \(T_2\): $$T_{2} 鈮� 11.86 \text{ years}.$$ Therefore, Jupiter's orbital period is approximately 11.86 years.

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